lsatcvat - Mathbox for Norm Megill

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Theorem lsatcvat 39043

Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 32315 analog.) (Contributed by NM, 10-Jan-2015.)

**Hypotheses**
Ref	Expression
lsatcvat.o	⊢ 0 = (0_g‘𝑊)
lsatcvat.s	⊢ 𝑆 = (LSubSp‘𝑊)
lsatcvat.p	⊢ ⊕ = (LSSum‘𝑊)
lsatcvat.a	⊢ 𝐴 = (LSAtoms‘𝑊)
lsatcvat.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lsatcvat.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lsatcvat.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)
lsatcvat.r	⊢ (𝜑 → 𝑅 ∈ 𝐴)
lsatcvat.n	⊢ (𝜑 → 𝑈 ≠ { 0 })
lsatcvat.l	⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))

**Assertion**
Ref	Expression
lsatcvat	⊢ (𝜑 → 𝑈 ∈ 𝐴)

**Proof of Theorem lsatcvat**
Step	Hyp	Ref	Expression
1		lsatcvat.o	. . 3 ⊢ 0 = (0_g‘𝑊)
2		lsatcvat.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lsatcvat.p	. . 3 ⊢ ⊕ = (LSSum‘𝑊)
4		lsatcvat.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑊)
5		lsatcvat.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec)
7		lsatcvat.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
9		lsatcvat.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
11		lsatcvat.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
13		lsatcvat.n	. . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 })
14	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 })
15		lsatcvat.l	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
16	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
17		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈)
18	1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17	lsatcvatlem 39042	. 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
19	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec)
20	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
21	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
22	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
23	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 })
24		lveclmod 21013	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
25	5, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod)
26		lmodabl 20815	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
27	25, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel)
28	2	lsssssubg 20864	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
29	25, 28	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
30	2, 4, 25, 9	lsatlssel 38990	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
31	29, 30	sseldd 3947	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
32	2, 4, 25, 11	lsatlssel 38990	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
33	29, 32	sseldd 3947	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
34	3	lsmcom 19788	. . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
35	27, 31, 33, 34	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
36	35	psseq2d 4059	. . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄)))
37	15, 36	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
38	37	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
39		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈)
40	1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39	lsatcvatlem 39042	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
41	29, 7	sseldd 3947	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
42	3	lsmlub 19594	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
43	31, 33, 41, 42	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
44		ssnpss 4069	. . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))
45	43, 44	biimtrdi 253	. . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)))
46	45	con2d 134	. . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈)))
47		ianor 983	. . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
48	46, 47	imbitrdi 251	. . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)))
49	15, 48	mpd 15	. 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
50	18, 40, 49	mpjaodan 960	1 ⊢ (𝜑 → 𝑈 ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ⊆ wss 3914 ⊊ wpss 3915 {csn 4589 ‘cfv 6511 (class class class)co 7387 0_gc0g 17402 SubGrpcsubg 19052 LSSumclsm 19564 Abelcabl 19711 LModclmod 20766 LSubSpclss 20837 LVecclvec 21009 LSAtomsclsa 38967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-0g 17404 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lvec 21010 df-lsatoms 38969 df-lcv 39012

This theorem is referenced by: lsatcvat2 39044

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